If you're from California, the biggest shock to you will not be the crowds or heat. I must warn you that it RAINS EVERY DAY in JUNE. TORRENTIAL DOWNPOURS WITH THUNDERSTORMS! EVERY DAY LIKE CLOCKWORK AT 4PM. Sometimes for only 20 minutes but other times for hours on the end.
Summer is the rainy season in Florida. You are going to be in for a big shocker.
I've been to California and seen people literally run screaming into buildings and cars when it was sprinkling. Literally full panic mode over rain so light that I wouldn't even bother with an umbrella if I had one. I also heard there that people call in sick to work and keep their kids home from school over a quarter of an inch of rain.
I can't imagine how they'd handle a Florida downpour. They usually get an inch in an hour at that time of the year and it comes down so hard and fast that you get soaked from head to toe if you stand out in it for even 30 seconds. I've also seen your other posts that you're not willing to do
Disneyland in the rain with your girls. Well, if you go to Disney World in June, you're going to have no choice but to deal with those afternoon thunderstorms.
I would suggest visiting during president's week in February or fall break in October. Crowds are about the same as early June but you are not going to get rain every day and might even get lucky enough to get no rain all week.
If you still plan on going in June, I'd suggest desensitizing your family to rain and teaching your girls not to be afraid of it. Buy them ponchos and have them practice walking around outside for a few minutes on the rare days it rains in California. Or you could keep your trip in June and plan on hitting the parks in the early mornings (it rarely rains before noon) and going back inside after lunch. Most of the Florida downpours are done by 7PM (but sometimes it can continue on and off through the night) so you can come back after dinner for night extra magic hours if your girls are able to handle staying up late.